About Fixed Points in CAT(0) Spaces under a Combined Structure of Two Self-Mappings
This paper investigates some fixed point-related questions including the sequence boundedness and convergence properties of mappings defined in spaces, which are parameterized by a scalar , where : are nonexpansive Lipschitz-continuous mappings and is a metric space which is a space.
A space, where is a real number related to the curvature, is a type of metric space where triangles of potential vertices being each set of three points are thinner, that is, of length being less than or equal to the corresponding so-called comparison triangles (namely, those whose sides have the same lengths as the sides of the original triangle) in the model spaces. The curvature in a space is bounded from above by . A particular case of spaces [1–3] is that arising when the curvature is bounded from above by 0 ( spaces. See, for instance, [4–10]). Complete spaces, being often referred to as Hadamard spaces (in honor to Jacques Hadamard), generalize Hilbert spaces to the nonlinear framework. In Hadamard spaces, there is a unique geodesic path joining each pair of given points. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. It is well-known that Hadamard spaces satisfy the following inequality:for each of the given points and some point , where is the unique midpoint of and , that is, , since Hadamard spaces are uniquely geodesic.
The study of geodesic paths is very relevant in spherical geometry, for instance, in the composition of trajectories through the Earth surface or in common planetary studies of distances. A metric space is a geodesic metric space if any two points can be joined by an arc length parameterized continuous curve (geodesic segment) whose length . It is well-known that a geodesic metric space is a space if every geodesic triangle in satisfies the inequality; namely, the distance between any two points of such a triangle is less than or equal to the distance between the corresponding points of the model triangle in the Euclidean space, that is, a triangle with sides of the same length as the sides of and then of the same perimeter. Therefore, the study of the metric properties of spaces has a major importance. In general, a geodesic metric space is a space if every geodesic triangle in with perimeter less than satisfies the inequality. The so-called spaces are of curvature and they are particular spaces of the most general spaces of curvature . To fix some general basic ideas, let us denote by the unique 2-dimensional Riemannian manifold with constant curvature of diameter being if and for . The spaces are those whose geodesic triangles , that is, those having geodesic segments as its sides, satisfy the so-called inequality; namely, the distances between points of are less than or equal to the distances between the corresponding points in their comparison triangles in the model space (i.e., those triangles whose sides have the same lengths as their counterparts as their corresponding -triangles). Note that if the metric space is a space, then there is a unique geodesic segment with joins and (with if ) [3, 11].
Furthermore, it is well-known that spaces are also spaces for and that - dimensional hyperbolic spaces with their usual metric are spaces and then spaces, whose simpler example is the -dimensional Euclidean space with its usual metric, so also spaces as well, whose simpler example is the unit sphere. Other relevant spaces are the so-called Euclidean buildings, which are abstract simplicial complexes, and the so-called cube complexes, [12–15]. The first ones give a systematic procedure for geometric interpretation of semisimple Lie groups and the study of semisimple groups over general fields, while the second ones are important, for instance, in the modelling process of robot trajectories on eventually irregular surfaces with eventual obstacles and predesigned admissible corridors for trajectory tracking.
This paper has two subsequent body sections. Section 2 gives some relating results for some of the various existing concepts of convexity in metric spaces such as -convexity, -convexity, midpoint convexity, convex structure, uniform convexity and near-uniform convexity, and Busemann curvature and its relation to convexity. Section 3 gives and proves some relevant properties related to uniform convexity and near-uniform convexity of geodesic metric spaces. It also studies mappings of the form parameterized by a scalar , defined by in a metric space , where are Lipschitz-continuous while not necessarily contractive mappings; that is, the Lipschitz constants are not necessarily less than unity. In particular, the convergence properties of sequences built from such mappings are formally studied and some conditions of existence of fixed points are given. Some illustrative examples are also presented and discussed. The main aim and motivation of the formal study of the sequences generated by such parameterized mappings, which are constructed with two generator mappings on a metric space , on spaces. The obtained and proved results rely on the boundedness and nonexpansive and contractive properties of the sequences generated by such mappings depending on the contractiveness/nonexpansiveness of both generator mappings. A potential application is the generation of admissible corridors whose extreme obstacle-free trajectories are defined by sequences generated by the two mentioned generator mappings. The generators of the corridor extreme trajectories are two primary mappings which define the studied parameterized mapping on the space. Those defined extreme trajectories can be bounded and/or convergent for each parameter value of the parameterized mapping of interest and could define the admissible trajectories of a robotic device or a movable body. One of the examples is concerned with this view on some potential applications.
Notation and denote the sets of integer and real numbers. . . . . is the closure of the set . is the closure of the convex hull of the family . denotes the set of fixed points of a mapping .
2. Some Preliminary Definitions and Results on Convexity, Uniform Convexity, and Curvature
Let be a complete metric space. It is said that it admits (nonnecessarily unique) midpoints if for any , there is a such that . Such a point is said to be a midpoint of and and is a geodesic space, [16–18].
Definition 1 (-convexity [16, 19]). Suppose a metric space which admits midpoints (or which has midpoints or which is midpoint convex). Then, is said to be -convex for some if, for each and each midpoint of and , For the case , the right-hand side of (2) is defined as a limit leading to . If is -convex, it is equivalently said to be ball convex, while if it is -convex it is equivalently said to be distance convex . is said to be strictly -convex for , if the inequality is strict for and strictly -convex if the inequality is strict for if .
Definition 1 leads to the direct conclusion below.
Assertion 2. If a metric space is midpoint convex, then it is -convex.
Assertion 3. If be -convex for some then, for any and each midpoint of and ,
Inequality (2) leads to the following direct result.
Assertion 4. If be -convex for some then, for any , each midpoint of and and each midpoint of and ,
Definition 5 (-Busemann curvature [16, 20]). Suppose a metric space which admits midpoints. Then, is said to satisfy the -Busemann curvature condition for some if, for each , each midpoint of and and each midpoint of and , one has
Assertion 6. Suppose a metric space which admits midpoints, with the midpoint map (or midset) being unique, and which satisfies the -Busemann curvature condition for some . Then, one hasfor any , where and are, respectively, the unique midpoints of and and and .
Proof. Since admits midpoints if , then the midpoint is unique since is unique , then . Thus, we can replace leading to an alternative right-hand side in (7) under the replacements and which when combined with (7) leads to (8).
Assertion 7. Assume that a metric space is midpoint convex (then it is 1-convex from Assertion 2) with unique midpoint map and that it satisfies the -Busemann curvature condition for some . Then, is -convex.
The following technical definitions are of interest to characterize near-uniform convexity.
Definition 8 (-separated family of points ). A family of points is -separated if .
Definition 9 (nearly uniformly convex space ). A -convex metric space is said to be nearly uniformly convex if, for any and for any -separated infinite family , with , and any such that , , there is some such that , where is the closure of the convex hull of the family .
It turns out that -convexity implies near-uniform convex but the converse is not true.
Definition 11 (see ). A convex metric space is said to be uniformly convex if, for any , there exists such that, for any and with and ,A uniformly convex metric space is also referred to commonly as uniformly 1-convex . This concept may be generalized as follows.
Definition 12. A convex metric space is said to be uniformly -convex if, for any , there exists such that, for any and any with and ,
Proposition 13. If a convex metric space is uniformly convex then it is uniformly -convex for any .
Proof. Since is uniformly convex then, for any , there exists such that (10) holds with any and any subject to and . Thus, . Choosing for any given and any given arbitrary , it follows for that so that is uniformly -convex.
Proposition 14. A convex metric space is nearly uniformly convex if, for any , there exists a strictly increasing such that, for any , any , and any -separated infinite family satisfying andfor some , where denotes an open ball of radius centred at .
Proof. Set for any given . Thus, if , one has from (12) thatsince since so that Since is trivially nonempty and then . Note also that, for any fixed , exists fulfilling (12). From Definition 9, is nearly uniformly convex.
Note that is nearly uniformly convex; then it is not necessarily uniformly convex since it can happen that, for some , any -separated infinite family satisfying and all , there exists some such that However, the converse is true as reflected in the next result.
Proposition 15. If a convex metric space is uniformly convex, then it is nearly uniformly convex.
Proposition 16. If is nearly uniformly convex and strictly -convex then, for any and for any -separated infinite family , with , and any such that , there is such that , and contains at most two points of such that with if and if with a choice .
Proof. Since is nearly uniformly convex and strictly -convex then any nonempty closed convex subset of is a Chebyshev set . Thus, for any , there are balls whose closures are Chebyshev sets consisting of at least two distinct points if their radius [16, 23] for any with . Since Chebyshev sets have a unique nearest neighbor in for each , , one has for that is a Chebyshev set of two unique elements one of them being its center . Then, one has since is nearly uniformly convex that for any , for any , with if with , if with .
3. Some Results on Contractiveness and Nonexpansiveness in CAT(0) Spaces
A metric space is (uniquely) geodesic if every two points of are joined by a unique geodesic segment which is the image of the geodesic path from to , that is, the isometry such that , and . A geodesic triangle consists of three vertices and three geodesic segments joining each pair of vertices. A model comparison triangle of the geodesic triangle is a triangle in the Euclidean space such that for . A geodesic metric space is a CAT space [4, 16, 21, 22] if ; (CAT inequality). See, for instance, [1, 4–8].
In the paper, we write for the unique such that and . Note that the midpoint of is .
Note the following from the basic results in Section 2.
(1) A geodesic space is a CAT space if and only if for any and all the following inequality is satisfied:(Proposition 1.1 ).
(2) A CAT space is uniformly -convex for any .
(4) A CAT space satisfies the -Busemann curvature condition for any from (7).
A general technical result involving constructions with two self-mappings in a space as follows.
Lemma 17. Let a metric space be a CAT(0) space and let the mapping be defined by for any and let be two self-mappings which satisfy the following conditions:for any given, some positive real constants and . Then, for any given and for any , the following properties hold:(i)(ii)
Proof. From (16) to (18), one gets:On the other hand, one has by completion of squares thatand from the triangle inequality for distances in (22a) and (22b) and the use of (18), one getsThe substitution of (23) into (22a) and (22b) with the use of (18) yieldsNow, the replacement of (18) and (24) into (21) leads towhich implies (19a)–(19c) and the proof of Property (i) is complete. On the other hand, one has directly from (19a)–(19c) after replacing , , , for that (20a)–(20c) holds for any as counterpart of (19a)–(19c) and Property (ii) is proved.
From Lemma 17(ii), we get the following result.
Lemma 18. Let a metric space be a CAT(0) space and consider the mapping defined in (17) via two self-mappings satisfying (18). Then, the following properties hold:
(i) Assume that (i.e., are both strictly contractive). Then, for any and .
(ii) Assume that(a)either (i.e., is strictly contractive), (i.e., is nonexpansive but noncontractive), and has a fixed point(b) (i.e., is strictly contractive), (i.e., is nonexpansive but noncontractive), and has a fixed point.Then,for any and .
(iii) If are both nonexpansive, where is a nonempty closed convex subset of , then for any and .
Proof. It follows for constants and that both and are strict contractions on so that one gets from (20a)–(20c) thatSince and are strict contractions on , and . Thus, for any and . Property (i) has been proved.
Now, assume that either and (then